-
Risk analysis of gravity dam instability using credibility
theory Monte Carlo simulation modelCao Xin1,2,3* and Gu
Chongshi1,2,3
BackgroundRisk analysis of gravity dam anti-sliding instability
involves factors of complicated uncer-tainty. In traditional
analysis method, main influencing factors of gravity dam
instabil-ity (e.g. various imposed loads, material properties and
geometric parameters) are all viewed as random variables (Peyras
et al. 2012). Now researchers gradually realize that fuzziness
is significant in risk analysis of gravity dam instability,
including: fuzzy stability failure limit state (Su et al.
2009), fuzzy mechanical parameters of dam foundation (Hua and Jian
2003), fuzzy safety monitoring data of dam and dam foundation (de
la Canal and Ferraris 2013; Gu et al. 2011) etc. Therefore,
problems of calculating precise failure risk ratio of gravity dam
instability influenced by complicated uncertainties and
deter-mining corresponding indexes and standards need to be
solved.
Since gravity dam involves factors of complicated uncertainty
during operation, researchers have applied possibility theory
(Zadeh 1965, 1996) into stability failure risk analysis of gravity
dam (Su and Wen 2013; Haghighi and Ayati 2016; Li et al.
2011;
Abstract Risk analysis of gravity dam stability involves
complicated uncertainty in many design parameters and measured
data. Stability failure risk ratio described jointly by
probabil-ity and possibility has deficiency in characterization of
influence of fuzzy factors and representation of the likelihood of
risk occurrence in practical engineering. In this arti-cle,
credibility theory is applied into stability failure risk analysis
of gravity dam. Stability of gravity dam is viewed as a hybrid
event considering both fuzziness and randomness of failure
criterion, design parameters and measured data. Credibility
distribution func-tion is conducted as a novel way to represent
uncertainty of influence factors of gravity dam stability. And
combining with Monte Carlo simulation, corresponding calculation
method and procedure are proposed. Based on a dam section, a
detailed application of the modeling approach on risk calculation
of both dam foundation and double sliding surfaces is provided. The
results show that, the present method is feasible to be applied on
analysis of stability failure risk for gravity dams. The risk
assessment obtained can reflect influence of both sorts of
uncertainty, and is suitable as an index value.
Keywords: Gravity dam, Anti-sliding stability, Risk analysis,
Fuzzy sets, Credibility theory
Open Access
© 2016 The Author(s). This article is distributed under the
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(http://creativecommons.org/licenses/by/4.0/), which permits
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RESEARCH
Xin and Chongshi SpringerPlus (2016) 5:778 DOI
10.1186/s40064-016-2508-7
*Correspondence: [email protected] 1 State Key Laboratory of
Hydrology and Water Resources and Hydraulic Engineering, Hohai
University, Nanjing 210098, Jiangsu, People’s Republic of ChinaFull
list of author information is available at the end of the
article
http://creativecommons.org/licenses/by/4.0/http://crossmark.crossref.org/dialog/?doi=10.1186/s40064-016-2508-7&domain=pdf
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Page 2 of 14Xin and Chongshi SpringerPlus (2016) 5:778
Sadeghi et al. 2010). Corresponding models have been
established to consider both ran-domness and fuzziness of influence
factors. However, in current studies, fuzziness and randomness of
factors influencing gravity dam stability are studied separately,
which makes the gained risk assessments are described jointly by
probability measure and pos-sibility measure. For example, a
certain failure risk probability interval is [R̃L, R̃U ] with
possibility α = 0.5. Given that probability and possibility are
independent, this descrip-tion fails to synthetically reflect the
contribution of fuzzy factors to risk assessments, and is difficult
to clearly characterize the likelihood of risk occurrence in
practical engineering.
Actually, fuzziness and randomness of influence factors have no
essential difference in stability failure risk analysis of gravity
dam. They shall be processed in the same frame. Therefore, this
article views gravity dam anti-sliding stability failure as a
hybrid event, uses credibility theory (Liu 2006) to consider both
randomness and fuzziness of failure criterion, design parameters
and measured data simultaneously and establishes credibility
stability failure risk analysis model of gravity dam to objectively
reflect both sorts of uncertainty. Combining present analysis model
with Monte Carlo simulation, calculation method and procedure are
proposed to analyze the risk of anti-sliding stability of gravity
dam.
Instability credibility risk ratio of gravity
damInstability risk calculation of gravity dam based on Monte Carlo
simulation can be divided into four steps: (1) determine the
instability model function of gravity dam; (2) identify and
quantify uncertainty of model factors; (3) simulate; (4) analyze
results and calculate instability risk ratio.
Instability risk calculation mode of gravity dam
Gravity dam instability is regarded as a random fuzzy event.
Fuzziness in the calculation model comes from model factors and
failure criterion. There will be three conditions: failure
criterion is determined while the model contains both random and
fuzzy factors; failure criterion is fuzzy and the model contains
random factors only; failure criterion is fuzzy and the model
contains both random and fuzzy factors.
Stability state function of gravity dam is expressed by Z and
fuzzy failure criterion is expressed by credibility density
distribution function η(Z). Then, credibility risk ratio model
considering both fuzzy failure criterion and fuzzy model factors
is:
where ϕ(Z) is credibility density distribution function of state
function and η(Z) is cred-ibility failure criterion. Through
appropriate conversion, failure probability model under rest two
conditions can be gained as following:
(1) State function is random fuzzy variable and failure
criterion is expressed by the limit state function. In other words,
when Z < K , the structure failed and the failure crite-
rion is η(Z) ={
1, ifZ < K0, ifZ ≥ K
. Then, the credibility risk ratio is degraded into:
(1)P̃ =∫ +∞
−∞
η(Z)ϕ(Z)dZ
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The state function contains random variables only and the
failure criterion is fuzzy. According to the integrated chance
theorem Ch{Θ × Y } = Pr{Y }, the credibility risk ratio degraded
into:
where f (Z) is random distribution density function of the state
function. Failure risk ratio is a real number expressed by
credibility distribution.
Since credibility failure criterion distribution function is
often sectionally derivable and derived functions are sectionally
continuous, it can get from partial integration of
Eq. (1):
where a and b are lower and upper limits of fuzzy interval of
failure criterion. Derived function of η(Z) is continuous in the
interval [a, b]. When Z < a, η(Z) = 1. When Z > a, η(Z) =
0.
Instability model function of gravity dam
Functional status of structure generally can be expressed by
performance function:
where Xi(i = 1, 2, . . . , n) is actions that could influence
the structure and environmental influence as well as performance
and geometric parameters of materials and rock soils.
For the simplest situation, Eq. (5) can be rewritten
into:
where R is resistance of structure and S is load effect of
structure.Main variables of instability risk ratio of gravity dam
foundation include upstream and
downstream water levels H1 and H2, silt elevation H3, uplift
pressure reduction coeffi-cient α1 and α2, shearing friction
coefficient f ′ and cohesion c′ of the contact surface between dam
concrete and dam foundation, volume weight of concrete γc and dam
pro-file size, and the force condition is shown in Fig. 1.
Shearing strength state function is:
In Fig. 1, W1, W2, W3 are upstream vertical water pressure,
dead weight of dam block and downstream vertical water pressure
respectively. P1, P2 and P3 are respectively upstream horizontal
water pressure, downstream horizontal water pressure and
hori-zontal silt pressure. U is uplift pressure. The calculation
expressions of these forces can be derived as follows:
(2)P̃(Z < K ) =∫ K
−∞
ϕ(Z)dZ = Φ(K ).
(3)P̃ =∫ +∞
−∞
η(Z)f (Z)dZ
(4)P̃ =∫ b
aΦ(Z)(−η̇(Z))dZ
(5)Z = g(X1,X2, . . . ,Xn)
(6)Z = R/S
(7)Z = R/S =(
f′ ∑
W + c′A)
/∑
P = g(
H1,H2,H3,α1,α2, f′, c′, γc,A
)
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Page 4 of 14Xin and Chongshi SpringerPlus (2016) 5:778
where γc, γw and γs are unit weight of concrete, water and silt
respectively.During deep anti-sliding stability analysis of both
surfaces of gravity dam, a deep weak
structural surface AB is chosen (Fig. 2), which is called
the main slip surface. BC is an assistant plane of fracture.
Stresses on ABD are dam weight (W ), total horizontal load of dam
(H), dead load of block ABC (G1), uplift pressure of AB (U1),
seepage pressure on the ABC–BCD interface (U3), acting force
between sliding blocks (Q). Stresses in BCD include opposite acting
force (Q), seepage pressure (U3), dead load of block BCD (G2), and
uplift pressure of BC (U2).
Shearing strength of ABD is:
Shearing strength of BCD is:
∑
W = W1 +W2 +W3 = γc
[
1
2D1(B− B0)+
1
2mB
2
0 +1
2nT
2
0
]
+ γw
(
H1B0 −1
2mB
2
0
)
+1
2γwnH
2
2
∑
P = P1 − P2 + P3 =
[(
H2
1 −H2
2
)
γw
]
/2+H23γs/2
U =1
2γw(H1B1 + α1H1B+ α2H2B+H2B2)
(8)
Z1 =f′
1
[
(W + G1)cosα0 −∑
Psin(ϕ − α0)− Qin(ϕ − α0)−U1 +U3sinα0]
+ c′
1A1
(W + G1)sinα0 +∑
Pcosα0 − U3cosα0 − Qcos(ϕ − α0)
(9)Z2 =f′
2[G2cosβ0 + Qsin(ϕ + β0)− U2 + U3sinα]+ c′
2A2
Qcos(ϕ + β0)− G2sinβ0 + U3cosβ0
Fig. 1 Anti-sliding stability analysis of gravity dam
foundation
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Page 5 of 14Xin and Chongshi SpringerPlus (2016) 5:778
where A1 is area of structural surface AB and A2 is area of
assistant plane of fracture BC. The calculation expressions can be
derived as follows:
For the loading condition shown in Fig. 2, ∑
W and ∑
P are the same as previously mentioned. The calculation
expressions of the rest can be derived as:
where b is width of dam block.According to equal safety
coefficient method, when ABC and BCD reach the limit
state at the same time, the double sliding surface model will
surface buckling failure. In other words, the limit state function
meets Z = Z1 = Z2. Based on this implicit func-tion, thrust Q and
the limit state function value Z can be calculated.
Risk identification of model factors
Uncertainty of model factors has to be analyzed when discussing
instability process of gravity dam using the opinion of mixed
uncertainty. Gravity dam, a complicated struc-tural system, will
suffer various concentrated forces and distributed forces (e.g.
dead load, hydraulic pressure and seepage pressure of dam
foundation) during construction and operation. Man causes of
gravity dam instability include flood, earthquakes, seepage of dam
foundation, material aging, and so on. The following text analyzes
uncertainty of water level and material parameters.
Water level is a random variable. Upstream water level is
related with pondage, res-ervoir inflow and discharge flow, while
downstream water level is mainly influenced by discharge flow.
Reservoir inflow is affected by rainfall and shows high
uncertainty. Discharge flow is controlled by hydraulic parameters
and also shows certain uncer-tainty. Additionally, hydrometry has
system and observation errors, which shows some
A1 = bT/cosα0
A2 = bT tanα0/sinβ0
G1 = γsbT2tanα/2
G2 = γsbcotβ(T tanα)2/2
Fig. 2 Anti-sliding stability analysis of double sliding
surfaces of gravity dam
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Page 6 of 14Xin and Chongshi SpringerPlus (2016) 5:778
uncertainty. Dam water level during the service period is
estimated through statisti-cal analysis of complete monitoring
sequences, which could use the classic frequency approach to
process it uncertainty. Upstream and downstream water levels can be
viewed as random variables (Salmon and Hartford 1995) which obey a
certain probabil-ity distribution.
Material parameters involved in the instability risk analysis of
gravity dam include vol-ume weights of dam and rocks as well as
shearing friction coefficient and cohesion of dam foundation and
slip surface. Volume weight of dam concrete was determined by test.
Sample size met requirement of statistical analysis and was treated
as a random variable. Uncertainty of mechanical parameters of rocks
in dam foundation includes: (1) uncertainty of engineering
geological investigation; (2) uncertainty of engineering
geo-logical rock group and rock structure. Fuzziness of mechanical
parameters of rocks, sta-tistical variables that combines survey
crew experiences and rock grouping, has more important
significance.
Uncertainty of failure criterion
Buckling failure of gravity dam is a progressive force when the
yield range expands from local to global. Some artificial
influences exist in understanding the failure criterion (Ma and Wu
2001). According to limit equilibrium state method, failure state
of sliding sur-face is divided clearly by the limit state function
Z = K . When safety coefficient of the sliding surface reaches a
specific value K , the whole system reaches the ultimate
equilib-rium state. However, fuzzy number is more practical to
express the progressive process. When considering fuzzy failure
criterion, fuzzy limit state of the structure is possibility
distribution that obeys to a certain membership function. For
example, the membership distribution function µ(Z) is a symmetric
triangular distribution:
According to definitions of credibility measure and credibility
distribution, credibility distribution density function
corresponding to the possibility distribution of failure cri-terion
is:
The symmetric triangular distributed fuzzy failure criterion is
turned into lower semi-trapezoid distributed credibility
distribution. This distribution converts from possibility
distribution interval into a univalent function of credibility
measure about limit state. Since credibility under limit state
drops significantly with the increase of state function Z, this
paper employed the following distribution form:
(10)µ(Z) =
2(Z−a)b−a
, for a ≤ Z ≤ a+b2
2(b−Z)b−a
, for a+b2
≤ Z ≤ b
0, for otherwise
(11)η(Z) =
1, for Z ≤ aZ−ab−a
, for a < Z ≤ b
0, for Z > b
(12)η(Z) =
1 for Z < a
ekZ−aZ−b for a ≤ Z < b
0 for Z ≥ b
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Page 7 of 14Xin and Chongshi SpringerPlus (2016) 5:778
where coefficients a, b and k could be determined according to
requirements on secu-rity level of different buildings and
corresponding standards based on expert experiences and information
entropy method (Shlyakhtenko 2005; Su et al. 2009)
(Figs. 3, 4, 5).
Random fuzzy simulation
In credibility stability failure analysis model of gravity dam,
some vari-ables (H1, H2, γc, α2) were vested with a random
distribution and others (
f ′, c′, α1, f′
1 , c′
1, f′
2 , c′
2, α, U1, U2, U3
)
were given with a possibility distribution. If the model
contains k random variables (X1, X2, . . . , Xk) and n− k fuzzy
variables (
Xk+1, Xk+2, . . . , Xn)
.
+
2
1
0
Fig. 3 Distribution of fuzzy failure criterion
1
+
2
0
0.5
Fig. 4 Lower semi-trapezoid distributed credibility distribution
of failure criterion
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Page 8 of 14Xin and Chongshi SpringerPlus (2016) 5:778
Since all fuzzy input variables are expressed in a fuzzy set,
model function was sim-ulated firstly under possibility measure.
Then, simulated results were converted into expression of
credibility distribution (Baudrit et al. 2005). Considering
independent ran-dom factors and fuzzy factors in the model, the
random fuzzy response of the random fuzzy limit state function of
gravity dam is gained through mixed algorithm (Baudrit et al.
2006), which is based on Monte Carlo simulation (Altarejos-García
et al. 2012):
1. Generate a random number in [0, 1] to every random variable.
The variable value which takes this random number as the
probability distribution is used as one sam-ple
(
px1 , px2 , . . . , pxk)
i.
2. Choose one α-cut and the membership function of the jth
sampling is recorded as µj.3. Generate maximum and minimum of the
response function under this horizontal
cut set, uij and lij .4. Choose another α-cut and repeat Step 2
and Step 3.5. Choose another random probability distribution
sequence and repeat Step 2–4.
For any random sample, the random fuzzy response expressed in
membership func-tion is gained through maximum and minimum values
of fuzzy response. In other words, for the ith (i = 1, 2, . . . ,N
) random sample
(
px1 , px2 , . . . , pxk)
i with a member-
ship of µZj(
j = 1, 2, . . . q)
, fuzzy response of state function is expressed as an interval
[
lij ,uij]
. Therefore, there are a total of N × q fuzzy intervals. Based
on this random fuzzy response and definition of credibility
measure, random fuzzy distribution Φi(Z) under the ith sample
expressed in credibility distribution can be gained as(Li and Liu
2009a):
where, Mi is arbitrary model function value under the ith random
sample, i = 1, 2, . . . ,N (Fig. 6).
(13)
Φi(Z) = Cr{M < Z} =1
2
(
maxMi
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Page 9 of 14Xin and Chongshi SpringerPlus (2016) 5:778
Post‑processing of simulated results
A series of model credibility distributions under different
random samples were gained through mixed algorithm. Based on
simulated results, the overall cumulative distribu-tion function
(CDF) of the stability failure model is calculated from
Eq. (14) (Li and Liu 2009b), which is recorded as Φ(Z):
where n(Z) is random sample size smaller than Z, N is total
random samples, A is arbi-trary interval of state function value
and y is credibility value of the interval A.
The overall instability risk ratio of gravity dam can be known
by bringing CDF into Eqs. (2)–(4). When failure criterion is
determined and the model function contains both random and fuzzy
variables, risk ratio P̃ is calculated from Eq. (2). When the
failure cri-terion is fuzzy and the model function only contain
random variables, risk ratio P̃ is cal-culated from Eq. (3).
When the failure criterion is fuzzy and the model function contains
both random and fuzzy variables, the overall risk ratio is
calculated from Eq. (4). Risk ratios are all determinate real
number on an interval [0, 1] (Fig. 7).
ApplicationBasic parameters
A serving gravity dam was chosen as the research object. It is a
first-grade structure with 100 years of design reference
period. The crest elevation, maximum height, normal pool level and
level of dead water are 384, 162, 173 and 370 m, respectively.
The downstream water level is basically stable. The upstream face
above the 295 m elevation is straight and the rest dam body
has 1:0.2 slope. The downstream face is vertical and crest width is
12 m. It was constructed on poor rock mass. There’s deep bed
rock and the dam founda-tion has high permeable rate. Two heavy
curtains were set on upstream and downstream
(14)Φ(Z) = max0≤α≤1,y≥α
(
α ∧minZ∈A,
n(Z)
N
)
/ ( )
1
0
0.5
( )
Fig. 6 Random fuzzy responses of the ith random sample expressed
in fuzzy set and credibility measure
-
Page 10 of 14Xin and Chongshi SpringerPlus (2016) 5:778
of the dam. The dam foundation is type-III rock mass. The
low-angle weak structural surface (T32-3, T32-5, JC2-1–JC2-8 etc.)
in the dam foundation inclined to the downstream and the low-angle
structural surface in downstream resisting foundation inclined to
the upstream form the main channel of deep foundation slippage. A
non-overflow dam sec-tion (elevation: 262 m) on the foundation
surface level was chosen in this paper. Rock grouping and soft rock
strata distribution in the dam foundation are shown in
Fig. 8.
Variables and statistical parameters or fuzzy parameters gained
from monitoring data and test data are listed in Tables 1 and
2.
Result analysisRandom fuzzy responses of the dam foundation
surface instability model and T2-5 deep instability model are shown
in Figs. 9 and 10.
Membership function of fuzzy failure state was determined.
Considering grade and design specifications of architecture,
parameters were determined 2.61, 3.08 and 1.05, respectively.
Results are shown in Table 3.
Distribution ofmodel parameters
Random fuzzy response ofmodel function
Limit state function ofgravity dam instability
Post-process of simulation
Credibility failure criterion ofgravity dam instability
Chance measure CDF ofgravity dam instability
Risk ratio of gravity dam instability
Dam foundation rock massdesign and monitoring data
Dam body design andmonitoring data
Fig. 7 Risk ratio calculation of gravity dam based on
credibility theory
Fig. 8 Rock grouping and soft rock strata distribution in the
typical foundation section
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Page 11 of 14Xin and Chongshi SpringerPlus (2016) 5:778
CDF of anti-sliding stability state function of dam foundation
and deep layers is shown in Fig. 11. It can be seen from
calculated results that instability risk ratios of both dam
founda-tion and deep layer are smaller than the
acceptable 1 × 10−4. Deep T32-3, JC2-6 and JC2-7
Table 1 Basic random variables and fuzzy variables
of dam body and foundation
Variables Variable type Parameter
Upstream depth H1(m) Random µH1 = 118, σH1 = 2.2
Downstream depth H2(m) Random µH1 = 8, σH1 = 0.8
Volume weight of dam concrete γc(
kN/m3)
Random µf ′ = 24.0, σf ′ = 0.5
Shearing friction coefficient of dam foundation surface f ′
Fuzzy [0.9, 1, 1, 1]
Shearing cohesion of dam foundation surface c′(MPa) Fuzzy [0.8,
0.9, 1.0]
Uplift pressure reduction coefficient of upstream curtain α1
Fuzzy [0.16, 0.20, 0.38]
Uplift pressure reduction coefficient of downstream curtain α2
Random µα2 = 0.30, σα2 = 0.03
Table 2 Basic parameters and fuzzy distribution parameters
of basement
Stratum Parameters of AB (c1/MPa)
Parameters of BC (c2/MPa)
Density γ (kN/m3)
α β
T32-3
f1 0.35 f2 [0.93, 0.98, 1.05] 26 [21°11′, 21°53′, 22°44′]
34°26′
c1 0.1 c2 [0.90, 1.00, 1.10]
T32-5
f1 0.35 f2 [0.93, 0.98, 1.05] 27 [21°35′, 23°26′, 25°21′]
35°15′
c1 0.1 c2 [0.90, 1.00, 1.10]
T32-6-1
f1 0.35 f2 [0.93, 0.98, 1.05] 27 [19°1′, 19°49′, 20°52′]
34°14′
c1 0.1 c2 [0.90, 1.00, 1.10]
JC2-3 f1 [0.35, 0.40, 0.44] f2 [0.93, 0.98, 1.05] 26 16°36′
29°5′
c1 [0.10, 0.12, 0.13] c2 [0.90, 1.00, 1.10]
JC2-4 f1 [0.35, 0.40, 0.44] f2 [0.93, 0.98, 1.05] 26 19°18′
33°1′
c1 [0.10, 0.12, 0.13] c2 [0.90, 1.00, 1.10]
JC2-5 f1 [0.35, 0.40, 0.44] f2 [0.93, 0.98, 1.05] 26 21°40′
34°19′
c1 [0.10, 0.12, 0.13] c2 [0.90, 1.00, 1.10]
JC2-6 f1 [0.35, 0.40, 0.44] f2 [0.93, 0.98, 1.05] 27 15°46′
38°51′
c1 [0.10, 0.12, 0.13] c2 [0.90, 1.00, 1.10]
JC2-7 f1 [0.35, 0.40, 0.44] f2 [0.93, 0.98, 1.05] 27 16°2′
36°12′
c1 [0.10, 0.12, 0.13] c2 [0.90, 1.00, 1.10]
2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
Z
Cum
ulat
ive
cred
ibili
ty v
alue
Fig. 9 Random fuzzy response to current value of instability
state function of dam foundation
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Page 12 of 14Xin and Chongshi SpringerPlus (2016) 5:778
have zero instability risk ratio. T32-5 lies in the fuzzy zone
which is about 0.2 m wide and has high fuzzy uncertainty, thus
resulting in its high risk ratio although it is deeply buried. The
most dangerous sliding surface is on JC2-4, with a risk ratio as
high as 6.61 × 10−5. This is caused by its low angle and
small burial depth as well as the giant upstream sliding block.
3.5 4 4.5 5 5.5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z
Cre
dibi
lity
cum
ulat
ive
valu
e
Fig. 10 Random fuzzy response to current value of T32-3 deep
instability state function of the slip surface
Table 3 Credibility instability risk ratios of dam
foundation and hypothetical sliding chan-nels
Italic value indicates the risk ratio of the most dangerous
sliding surface
Slip surface P̃
Dam base 2.25 × 10−5
JC2-3 5.64 × 10−6
T32-6-1 2.68 × 10−5
JC2-4 6.61 × 10−5
JC2-5 3.42 × 10−5
T32-5 5.08 × 10−6
JC2-6 0
JC2-7 0
T32-3 0
(a) (b)
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z
Proba
bility
2.8 3 3.2 3.4 3.6 3.8 4 4.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z
Proba
bility
Fig. 11 Cumulative distribution of anti-sliding instability
probability, a JC2-4, b dam foundation
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Page 13 of 14Xin and Chongshi SpringerPlus (2016) 5:778
To analyze effect of fuzzy factor f and c change on risk ratio
Pr, sensitivity analysis was implemented by using the following
method. Firstly, calculate risk ratio under different variances of
f while distribution of c and mean of f are fixed. Draw the curve
Pr ∼ Vf . Similarly, draw curve Pr ∼ Vc. Results of sensitivity
analysis of stability failure risk assessment of dam foundation are
shown in Fig. 12. When variance of fuzzy variable increases,
risk ratio of the system increases. This conforms to the general
rule.
ConclusionsStability failure of gravity dam involves complicated
uncertainty. In this article, after adequately considering
uncertainty of various factors, a method to analyze stability
fail-ure risk of gravity dam and corresponding calculation
procedure are proposed. Conclu-sions are drawn as follows.
To overcome the deficiency of existing risk analysis method that
considers randomness and fuzziness separately, this article applies
the credibility theory into dam failure and establishes a
credibility stability failure risk analysis model of gravity dam to
integrate ran-domness and fuzziness together. Based on Monte Carlo
simulation, a mixed algorithm is combined with post-processing of
credibility risk analysis mode. And general calculating method is
adopted. Stability failure of gravity dam are viewed as hybrid
events. Inputs of this calculating method are a series of
probability distribution and fuzzy sets. By introduc-ing
credibility measure, fuzziness in hybrid variables is mapped from
possibility space into the probability space, so risk ratio
obtained is represented by a determinate real number.
The example demonstrates that the present method is effective
for providing decision support to safety assessment of fuzzy
stability of gravity dam. Sensibility analysis results show that
credibility risk ratio can sensitively reflect variance change of
fuzzy factors. Risk ratio described by credibility measure reflects
both randomness and fuzziness and agrees with description habit of
traditional probability risk ratio. If there are indexes and
standards on credibility risk ratio, credibility theory could
become an effective represen-tation tool of fuzzy-random stability
failure risk analysis of gravity dam.
0 1 2 3 4
x 10-3
1
1.5
2
2.5x 10
-4
Varience
Ris
k pr
obab
ility
~ ’
~ ‘
Fig. 12 Relationship between risk ratio of dam foundation Pr ~
variance of factors f ′ and c′
-
Page 14 of 14Xin and Chongshi SpringerPlus (2016) 5:778
Authors’ contributionsCX designed the research, drafted the
manuscript, reviewed the literature, prepared the Matlab coding for
CTMCS model and analyzed the data. GC provided guidance, conducted
final editing and proofreading. Both authors read and approved the
final manuscript.
Author details1 State Key Laboratory of Hydrology and Water
Resources and Hydraulic Engineering, Hohai University, Nanjing
210098, Jiangsu, People’s Republic of China. 2 National Engineering
Research Center of Water Resources Efficient Utilization and
Engineering Safety, Hohai University, Nanjing 210098, Jiangsu,
People’s Republic of China. 3 College of Water Con-servancy and
Hydropower Engineering, Hohai University, Nanjing 210098, Jiangsu,
People’s Republic of China.
AcknowledgementsThis research has been partially supported by
National Natural Science Foundation of China (SN: 51139001,
51479054) and Research Fund for the Doctoral Program of Higher
Education of China (SN: 20020294005, 20120094130003).
Competing interestsThe authors declare that they have no
competing interests.
Received: 19 January 2016 Accepted: 2 June 2016
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Risk analysis of gravity dam instability using credibility
theory Monte Carlo simulation modelAbstract BackgroundInstability
credibility risk ratio of gravity damInstability risk
calculation mode of gravity damInstability model function
of gravity damRisk identification of model
factorsUncertainty of failure criterionRandom fuzzy
simulationPost-processing of simulated results
ApplicationBasic parameters
Result analysisConclusionsAuthors’ contributionsReferences
